So I was going to start a new Skyrim game, and since I would be using multiple computers in different places, I wanted to be able to use the Steam Cloud. However, I know its filled with a bunch of garbage saves from my old characters and I really don't want any of those cluttering up my Skyrim. Is there any way to get rid of those old Steam cloud saves permanently or at least make sure they don't keep re-appearing?

Yes.

In the Steam Library area, right click on Skyrim -> Select Properties -> In the window that pops up, search the various tabs for the Steam Cloud option and disable it.